Questions tagged [bivariate-distributions]
For questions on bivariate distributions, the combined probability distribution of two randomly different variables.
0
votes
1answer
21 views
Conditional distribution for two random variables
I recently came across and exercise from a past exam and I was wondering how to solve it.
Two independent random variables $A$ and $B$ are given and they both follow the exponential distribution but ...
2
votes
0answers
38 views
Find the marginal pdf of $X$ and $Y$
Suppose that $(U,V)$ has the following joint pdf:
$$f_{U,V}(u,v)=\exp(-\theta u-\theta^{-1}v)$$
, where $u\geq0$, $v\geq0$, $\theta>0$. Define $X=UV$ and $Y=U/V$. Find the marginal pdf of $X$ and $...
0
votes
1answer
32 views
Bivariate Pareto accumulated distribution function from its density function
I am trying to check how a density function of a bivariate Pareto distribution f(x,y) goes to its accumulated function F(x,y):
$$ f(x,y) = \frac{a(a+1)}{b_1b_2}\biggl [-1 + \frac x {b_1} + \frac y{...
0
votes
1answer
48 views
Is it a distribution function?
Consider $F(x,y) = \mathbb{I}[x+y \geq 1] $ .
$\mathbb{I} $ denotes the indicator function . So , $F(x,y)$ is 1 iff $x+y \geq 1$ and $0$ otherwise .
Can we consider to be a bivariate distribution ...
2
votes
1answer
58 views
Finding $L_n$ so that $\lim_{n \rightarrow \infty} P(L_n<\rho<1)=\alpha$
Let $(X_1,Y_1),(X_2,Y_2),..,(X_n,Y_n)$ be iid pairs of random variables with $E(X_1)=E(Y_1)$, $\text{Var}(X_1)=\text{Var}(Y_1)=1$,and $\text{cov}(X_1,Y_1)=\rho \in(-1,1)$.
Given $\alpha>0$ , obtain ...
0
votes
0answers
39 views
Derivatives for box integral of a bivariate normal distribution
I'm having quite a bit of trouble trying to understand how to to calculate partial derivatives of a specific function.
Suppose I have the standard bivariate normal density function:
$$f(x,y,\rho)=\...
0
votes
0answers
33 views
What is the probability that both roots of the equation Ax2+Bx+C=0 are real?
Let $A, B,$ and $C$ be independent random variables, uniformly distributed over $[0,10], [0,6],$ and $[0,12]$ respectively. What is the probability that both roots of the equation $Ax^2+Bx+C=0$ are ...
0
votes
0answers
6 views
Contours in the bivariate graph connect regions of approximately equal popula-tions. Which of the following interpretations is correct?
Contours in the bivariate (weight, height) graph connect regions of approximately equal popula-tions. Which of the following interpretations is correct?
I can not understand anything about this ...
0
votes
1answer
74 views
Find the PDF for $U=\frac{Y_1}{Y_2}$
I need some help on the following problem:
Let $Y_1$ and $Y_2$ be two random variables with the following density function:$$f_1(y_1)=
\begin{cases}
6y_1(1-y_1), & \text{if } 0\le y_1\le 1 \\...
0
votes
0answers
13 views
Difference between inverse CDFs in a bivariate RV
X and Y are normal distributions with different mean and variance. They are jointly distributed with some correlation $\rho$ (assume they are simply a bivariate normal RV).
Let $F_X(x)$ and $F_Y(x)$ ...
0
votes
0answers
49 views
Spearman's Correlation Coefficient for Bivariate Normal Distribution
Referring to the answer here https://stats.stackexchange.com/a/66617
It is written that $\rho_s(X_1,X_2) = \rho(F_1(X_1),F_2(X_2))$
My Questions are :-
Is that forumla correct? Because I am not ...
0
votes
0answers
17 views
Bound on probability of bi-variate chi-square distribtuion
If I have two random variables $u_1 = x_1^2$ and $u_2 = x_2^2$ where $x_1,x_2 \sim \mathcal{N}(0,1)$ and $E(x_1x_2) = \rho$. The what is the pdf or bound on the pdf of $P(u_1<c,u_2>c)$?
Edit: ...
0
votes
0answers
14 views
Variance of truncated 2d Gaussian
To find the expectation of the Truncated Gaussian E($z_1^2$| $z_1^2 \leq \tau , z_2^2 \geq \tau$). Where $\boldsymbol{z} = [z_1,z_2]^T$ and $\boldsymbol{z} \sim \mathcal{N}(\boldsymbol{0},C)$, where $...
0
votes
0answers
184 views
MLE of Parameters of Bivariate Normal Distribution
I am working through find the maximum likelihood estimators of the bivariate normal distribution, without using matrices. I have the following density function:
$f(Y_1,Y_2) = \frac{1}{2\pi\sigma_1\...
0
votes
0answers
22 views
Calculate a 2-dimensional Gauss-Hermite quadrature approximation
I need to calculate an integral in which the second term is a bivariate normal density.
I thought about using a 2-dimensional Gauss-Hermite quadrature.
But not being familiar with the subject, I do ...
1
vote
1answer
63 views
Linear combination of normal distribution which is not normal
Let $\xi_1, \xi_2$ be i.i.d N(0,1).
Define
$(X_1,X_2)=\begin{cases}
(\xi_1, |\xi_2|) \quad \xi_1 \geq 0 \\
(\xi_1,-|\xi_2|) \quad \xi_1 < 0
\end{cases}$
This means we can rewrite $X_1=\xi_1$ and ...
1
vote
2answers
53 views
Proving $(X,Y)$ is a normal vector when $X\sim N(1,1)$ and $Y\mid X\sim N(3X,4)$
Suppose I have a random vector $(X,Y)$ with $X\sim\mathcal{N}(1,1)$ and $Y|X = x \sim\mathcal{N}(3x,4)$.
I need to prove that $(X,Y)$ is a normal vector as well.
To do that I want to explicitly ...
0
votes
1answer
38 views
Find Distribution of Sum of Random Variables given only Joint Distribution
So if we have two random variables $X, Y$, with unknown distributions. Then we have a random variable $Z$, such that $Z=X+Y$.
Firstly, how do we find the CDF of $Z$, i.e. $F_Z(z)$, given the joint ...
0
votes
1answer
36 views
Correlation Coefficient r, Formula Explained Intuitively
I've seen several videos, Khan Academy included, explaining the correlation coefficient formula but none explain the "logic" behind the formula, not to my satisfaction anyways.
The Formula: ...
0
votes
0answers
27 views
Expectation computation of correlated normal random variables
I'm struggling to prove the following maybe some of you can help me. Here is my problem
Let $\mathbf{X}\sim\mathcal{N}(\mu,\Sigma)$ where $\mathbf{X}=\pmatrix{X_1\\X_2}, \mu=\pmatrix{\mu_1\\\mu_2}$ ...
1
vote
1answer
38 views
Find the marginal distribution of an point randomly chosen on an ellipse
This exercise comes from rice 3.6 and states: A point is chosen randomly in the interior of an ellipse:
$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
Find the marginal densities of the $x$ and $y$ ...
0
votes
1answer
28 views
Bivariate normal: Expected value and variance
I have this simple exercise to do and I'm new to this topic.
Seeing the slides of my professor, I would solve the problem in this way:
$E(Y)=c_1 μ_1+c_2 μ_2$
$E(Y)=-54.2424$
$Var(Y)=σ_{22}$
$Var(...
0
votes
0answers
103 views
Find the expected value and standard deviation of the returns for the following portfolio
Suppose you want a portfolio composed of AT&T, Cigna, Disney, and Ford. Find the expected value and standard deviation of the returns for the following portfolio
I know that I have to use
...
1
vote
1answer
20 views
Bivariate Normal Proof
In our discussion of the bivariate normal, there is
an expression for $E(Y|X = x)$.
a. By reversing the roles of X and Y give a similar
formula for $E(X|Y = y)$.
b. Both $E(Y|X = x)$ and $E(X|Y = y) ...
0
votes
1answer
83 views
Conditional marginal distribution of conditional bivariate normal distribution
I have a bivariate normal distribution$$(X, Y)\sim N(\mu_{x}, \mu_{y}, \sigma_{x}^2, \sigma_{y}^2, \rho)$$ My question is : when $X > k$ ($k$ is a constant),how to get the distribution of $Y$?
Can ...
0
votes
0answers
11 views
conditional and marginal of bivariate t distribution - i.i.d or not?
I have a bi variate t-distribution with $X=(X_1, X_2)=t(\mu, \sigma^2, v)$
$$\mu=(1,2)',\sigma^2=\left(\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right), v=4
$$
How can I calculate the ...
1
vote
1answer
50 views
Finding the Probability for a Bivariate Normal Distribution [duplicate]
Given $(x_1,x_2)' \sim N_2 \left(\bf{0},\Sigma\right) = N_2 \left(\left(\begin{array}{l}0\\0\end{array}\right), \left(\begin{array}{l}1&\rho\\\rho&1\end{array}\right)\right)$, find $Pr(x_1>...
0
votes
1answer
37 views
Calculate the joint distribution of $X, Y$ and $Z$, following a bivariate normal distribution
Let $X$ and $Y$ follow a bivariate normal distribution with $\mu = (0,0)^T$, $\sigma_x=1$, $\sigma_y=1$ and correlation $\rho =0.5$. Also, suppose that a pair $Z$ and $Y$ follow a bivariate normal ...
-3
votes
2answers
78 views
Find the distribution of $Y-X$. [closed]
Let $X \sim N(0,1)$ and $Y \sim N(0,2)$ respectively, independently of each other. Find the distribution of $Y-X$.
How do I proceed? As I am a beginner I don't know how to approach. Please help me in ...
0
votes
0answers
10 views
Formula for copulas of bivariate mixed (discrete and continuous) data
I have a question as I am new to copulas. I have bivariate data (X,Y), one is discrete and one is continuous distributed. Can I use the usual formula for the common density function given by
$$f_X(x) \...
1
vote
0answers
25 views
Is taking $3\sigma$ limits for bivariate correlated random variables as the confidence limits an under estimation?
If the two random variables are uncorrelated, $3\sigma$ limit would mean that the random variables lie inside a hyperrectangle with center coordinates being means of random variables. But if two ...
0
votes
2answers
48 views
Bivariate probability distribution of a pdf
Given
$$f ( x_1 , x_2 ) = \begin{cases}6 ( 1 - x_2 ) &,&0 ≤ x_1 ≤ x_2 ≤ 1
\\ 0 &&\text{ Otherwise.}\end{cases}$$
Find $P ( X_1 ≤ 0.75, X_2 ≥ 0.5 )$
The correct answer is $31/64$.
...
1
vote
0answers
56 views
Circle-shaped integration of a bivariate normal distribution
I have a bivariate Normal Distribution of positions in a 2D space given by the marginal distributions of x and y components: $N_x(\mu_x,\sigma_x)$, $N_y(\mu_y,\sigma_y)$.
I want to determine the ...
0
votes
1answer
18 views
Checking the validity of Bivariate Distribution function
Which of the following is /are bivariate Distribution function:
$F(x,y)=\begin{cases}
0 & \text{ if } x+y<0\\
1 & \text{ if } x+y\geq0
\end{cases}$
$F(x,y)=\begin{cases}
1-e^{-...
2
votes
2answers
40 views
calculate $\operatorname{cov}(X,Y)$ from $f_{x,y}(x,y)$
I have the following density function:
$$f_{x, y}(x, y) = \begin{cases}2 & 0\leq x\leq y \leq 1\\ 0 & \text{otherwise}\end{cases}$$
We know that $\operatorname{cov}(X,Y) = E[(Y - EY)(X - EX)]$...
0
votes
1answer
24 views
Calculate probability $P(X_1\geq1/2, X_2\geq1/2)$ for $F_{X_1,X_2}$
I am studying for an exam and came across this example to calculate probability $P(X_1\geq1/2, X_2\geq1/2)$ for $F_{X_1,X_2}$ = $1/2x_1x_2(x_2^2+x_1^2)$ if $0\leq x_1\leq 1,0\leq x_2\leq 1$.
The ...
1
vote
1answer
120 views
Distribution of $Z=\frac{UX+VY}{\sqrt{U^2+2\rho UV+V^2}}$ when $(X,Y)$ is bivariate normal and $U,V$ independent of $X$ and $Y$
$X$ and $Y$ have bivariate normal distribution with zero means, unit variances, and correlation $\rho$. Let $U$ and $V$ be independent of $X$ and $Y$.
How can we find the distribution of $Z = \...
1
vote
0answers
38 views
Conditional normality implies marginal normality or bivariate normality? [closed]
Let X and Y be two random variables.
If we know that both Y|X and X|Y are normal, then can we conclude that
(1) X and Y are normal respectively;
(2) X and Y are bivariate normal?
Thanks!
Edited:
...
-1
votes
1answer
140 views
Formula for expectation of Bivariate Data [closed]
`Suppose that $(X, Y )$ has a uniform distribution on the parallelogram with vertices at
$(0,0)$
$(1292,1000)$
$(1526,0)$
$(2818,1000)$
Calculate the means of $X$ and of $Y$.
I don't know the ...
0
votes
1answer
22 views
Bivariate distribution of the discrete type question
A homework questions asks:
John has two quarters and two dimes. He tosses these four coins 3 times. Let U be the number of heads of quarters, and V be the number of heads of dimes.
Denote X = U +...
1
vote
2answers
51 views
Non-normal Bivariate distribution with normal margins
I was working on a question and it stated the following:
Now I am supposed to show that the bivariate distribution is non normal even though the marginal distributions are. I generated 1000 iid ...
-2
votes
1answer
49 views
Joint normal and exponential
if X has exponential exp(lambda)and Y has normal distribution N(0,1) . X and Y are independent. how can one find p(X < Y)?
My thoughts were to integrate :
$$\int_{-\infty}^{+\infty} F_X(y)f_Y(y) \...
1
vote
1answer
273 views
Are uncorrelated linear combinations of the elements of a multivariate normal distribution always independent of each other?
To make a simple example, we could have $X \sim N_2 \left( \begin{pmatrix} 0 \\ 0\end{pmatrix}, \begin{pmatrix} 4 & -4 \\ -4 & 2\end{pmatrix} \right)$ which gives $X_1 \sim N(0, 4)$ and $X_2 \...
1
vote
0answers
52 views
Is the joint distribution of any two uncorrelated normally distributed random variables bivariate normal?
Originally I was only going to ask if the joint distribution of any two normally distributed is random variables bivariate normal, however it seems that a large set of counterexamples can be found ...
1
vote
1answer
29 views
Covariance matrix of $(\bar{X}, \bar{X^2})$
I found a pdf that claims that asymptotically:
$(\frac{\sum X_i}{N}, \frac{\sum X_i^{2}}{N})$ converges in distribution to a bivariate random variable with mean $(\mu_1,\mu_2)$ and covariance matrix $...
0
votes
1answer
500 views
Expectation for Trinomial distribution
I am trying to understand the proof of $\mathrm{E}[xy]=n(n-1)p_1p_2$
where x,y have a trinomial distribution with pmf:
$p(x,y) = \frac{n!}{x!y!(n−x−y)!}p_1^xp_2^yp_3^{n−x−y}$
The proof has the ...
0
votes
0answers
33 views
What is wrong with the following approach of obtaining the joint distribution function.
Let the joint probability density function of $X,Y$ is given as:
$f(x,y)=\frac{1}{4}[1+x^3y^3]$ where $-1\leq x \leq 1$ and $-1\leq y \leq 1$. It is required to obtain the joint probability density ...
6
votes
1answer
318 views
Variance of Z = max(X,Y) where X Y are jointly bivariate normal
I have a question about the bivariate normal r.v.'s
Given $X, Y \sim \operatorname{Normal}(0,1)$ with correlation coefficient $\rho$. Let $Z=\max(X,Y)$. Show that $\operatorname E Z^2=1$.
My ...
1
vote
2answers
58 views
Find the density function of the sum $(X,X+Y)$.
Problem
I have to prove the following:
Let $X$ and $Y$ be independent continuous random variables with density function $f:\mathbb R\to\mathbb R$. \
Prove that $(X,X+Y)$ is a continuous bivariate ...
0
votes
1answer
720 views
Finding the conditional probability given the joint probability density function
f(x,y) = e^-x if 0 <= y <=x < \infty
i need to find the P(X<3|Y<2) & P(X<3|Y=2).
i'm struggling with the first probability. i'm not sure how to evaluate the conditional given ...
